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certify_rational.c
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/*
* Cadenza
*
* Jonathan Hauenstein <[email protected]>
* Alan Liddell <[email protected]>
* Ian Haywood <[email protected]>
*
* certify_rational.c: Rational-valued functions related to certification
*/
#include "cadenza.h"
/*********************************************************************
* takes X^2, Y^2, Z^2, returns 1 if X + Y <= Z (where X, Y, Z >= 0) *
*********************************************************************/
int lte_sqr_rational(mpq_t X_sqr, mpq_t Y_sqr, mpq_t Z_sqr) {
/* first inequality: X^2 + Y^2 - Z^2 <= 0 */
mpq_t first;
mpq_init(first);
mpq_add(first, X_sqr, Y_sqr);
mpq_sub(first, first, Z_sqr);
/* fails */
if (mpq_cmp_ui(first, 0, 1) >= 0) {
mpq_clear(first);
return 0;
}
mpq_clear(first);
/* second inequality: (Z^2 - (X^2 + Y^2))^2 >= 4X^2Y^2 */
mpq_t second_left;
mpq_t second_right;
mpq_t four;
mpq_init(second_left);
mpq_init(second_right);
mpq_init(four);
mpq_set_ui(four, 4, 1);
mpq_add(second_left, X_sqr, Y_sqr);
mpq_sub(second_left, Z_sqr, second_left); /* Z^2 - (X^2 + Y^2) */
mpq_mul(second_left, second_left, second_left); /* (Z^2 - (X^2 + Y^2))^2 */
mpq_mul(second_right, X_sqr, Y_sqr); /* X^2Y^2 */
mpq_mul(second_right, four, second_right); /* 4X^2Y^2 */
/* succeeds */
if (mpq_cmp(second_left, second_right) >= 0) {
mpq_clear(second_left);
mpq_clear(second_right);
mpq_clear(four);
return 1;
}
mpq_clear(second_left);
mpq_clear(second_right);
mpq_clear(four);
return 0;
}
/*****************************
* compute ||J(f)^-1 * v||^2 *
*****************************/
void compute_norm_sqr_Jv_rational(polynomial_system *F, rational_complex_vector v, rational_complex_vector x, mpq_t *norm_sqr_Jv) {
int retval = 0, *rowswaps = NULL;
rational_complex_vector f;
/* X = J^-1 * v */
rational_complex_vector X;
rational_complex_matrix J;
rational_complex_matrix LU;
initialize_rational_vector(f, 0);
initialize_rational_vector(X, 0);
initialize_rational_matrix(J, 0, 0);
initialize_rational_matrix(LU, 0, 0);
eval_polynomial_system_rational(f, J, F, x);
retval = LUdecomp_rational(LU, &rowswaps, J);
/* success */
if (!retval) {
LUsolve_rational_vector(X, LU, rowswaps, v);
norm_sqr_rational_vector(*norm_sqr_Jv, X);
} else {
mpq_set_ui(*norm_sqr_Jv, 0, 1); /* this may need to be something else */
}
clear_rational_vector(f);
clear_rational_vector(X);
clear_rational_matrix(J);
clear_rational_matrix(LU);
free(rowswaps);
rowswaps = NULL;
}
/*********************************************
* return 1 if interval certifies continuous *
*********************************************/
int is_continuous_rational(rational_complex_vector v, mpq_t t_left, mpq_t t_right, rational_complex_vector x_left, rational_complex_vector x_right, polynomial_system F_left, polynomial_system F_right, mpq_t alpha_sqr_left, mpq_t alpha_sqr_right, mpq_t gamma_sqr_left, mpq_t gamma_sqr_right) {
int retval = 0, left_success = 0, right_success = 0;
mpq_t alpha_star_sqr;
mpq_t t_dist_sqr;
mpq_init(alpha_star_sqr);
mpq_init(t_dist_sqr);
mpq_set_ui(alpha_star_sqr, 4, 100);
mpq_mul(alpha_star_sqr, alpha_star_sqr, alpha_star_sqr);
mpq_sub(t_dist_sqr, t_right, t_left); /* t_2 -t_1 */
mpq_mul(t_dist_sqr, t_dist_sqr, t_dist_sqr); /* |t_2 - t_1|^2 */
/* we want either of magic_number_left or _right to be less than alpha_star */
mpq_t magic_number_left;
mpq_t norm_sqr_Jv_left;
mpq_t magic_number_right;
mpq_t norm_sqr_Jv_right;
mpq_init(magic_number_left);
mpq_init(norm_sqr_Jv_left);
mpq_init(magic_number_right);
mpq_init(norm_sqr_Jv_right);
compute_norm_sqr_Jv_rational(&F_left, v, x_left, &norm_sqr_Jv_left);
compute_norm_sqr_Jv_rational(&F_right, v, x_right, &norm_sqr_Jv_right);
/* ||J^-1 * v||^2 */
mpq_set(magic_number_left, norm_sqr_Jv_left);
mpq_set(magic_number_right, norm_sqr_Jv_right);
/* ||J^-1 * v||^2 * g^2 */
mpq_mul(magic_number_left, magic_number_left, gamma_sqr_left);
mpq_mul(magic_number_right, magic_number_right, gamma_sqr_right);
/* ||J^-1 * v||^2 * g^2 * |t_1 - t_2|^2*/
mpq_mul(magic_number_left, magic_number_left, t_dist_sqr);
mpq_mul(magic_number_right, magic_number_right, t_dist_sqr);
/* run the tests */
left_success = lte_sqr_rational(alpha_sqr_left, magic_number_left, alpha_star_sqr);
right_success = lte_sqr_rational(alpha_sqr_right, magic_number_right, alpha_star_sqr);
/* test fails for both left and right */
if (left_success == 0 && right_success == 0)
retval = 0;
else {
mpq_t gamma_sqr_inv;
mpq_t points_dist_sqr;
mpq_t magic_number_sqr;
mpq_init(gamma_sqr_inv);
mpq_init(points_dist_sqr);
mpq_init(magic_number_sqr);
rational_complex_vector points_dist;
initialize_rational_vector(points_dist, x_left->size);
/* test succeeds for both left and right */
if (left_success && right_success) {
/* singularities segfault on the Mac */
mpz_t denom_left;
mpz_t denom_right;
mpz_init(denom_left);
mpz_init(denom_right);
mpq_get_den(denom_left, gamma_sqr_left);
mpq_get_den(denom_right, gamma_sqr_right);
if (mpz_cmp_ui(denom_left, 0) == 0) {
if (mpz_cmp_ui(denom_right, 0) == 0)
mpq_set(gamma_sqr_inv, gamma_sqr_left); /* might as well pick left since both are singularities */
else
mpq_set(gamma_sqr_inv, gamma_sqr_left); /* gamma_sqr_right wins without a fight */
} else if (mpz_cmp_ui(denom_right, 0) == 0) {
mpq_set(gamma_sqr_inv, gamma_sqr_left); /* gamma_sqr_left wins without a fight */
} else {
mpq_set_min(gamma_sqr_inv, gamma_sqr_left, gamma_sqr_right); /* choose the lowest value for (both finite) gamma */
}
mpq_inv(gamma_sqr_inv, gamma_sqr_inv); /* set gamma to 1/gamma */
} else if (left_success) { /* test succeeds for left only */
mpq_inv(gamma_sqr_inv, gamma_sqr_left);
} else { /* test succeeds for right only */
mpq_inv(gamma_sqr_inv, gamma_sqr_right);
}
/* magic number is now to be 79/(1000*gamma) */
mpq_set_ui(magic_number_sqr, 79, 1000);
mpq_mul(magic_number_sqr, magic_number_sqr, magic_number_sqr); /* (79 / 1000)^2 */
mpq_mul(magic_number_sqr, magic_number_sqr, gamma_sqr_inv); /* (79 / (1000 * gamma))^2 */
subtract_rational_vector(points_dist, x_left, x_right);
norm_sqr_rational_vector(points_dist_sqr, points_dist); /* ||x_1 - x_2||^2 */
if (mpq_cmp(points_dist_sqr, magic_number_sqr) <= 0) {
retval = 1;
} else {
retval = 0;
}
mpq_clear(points_dist_sqr);
mpq_clear(magic_number_sqr);
mpq_clear(gamma_sqr_inv);
clear_rational_vector(points_dist);
}
mpq_clear(alpha_star_sqr);
mpq_clear(t_dist_sqr);
mpq_clear(magic_number_left);
mpq_clear(magic_number_right);
mpq_clear(norm_sqr_Jv_left);
mpq_clear(norm_sqr_Jv_right);
return retval;
}
/************************************************
* return 1 if interval certifies discontinuous *
************************************************/
int is_not_continuous_rational(rational_complex_vector v, mpq_t t_left, mpq_t t_right, rational_complex_vector x_left, rational_complex_vector x_right, polynomial_system F_left, polynomial_system F_right, mpq_t alpha_sqr_left, mpq_t alpha_sqr_right, mpq_t gamma_sqr_left, mpq_t gamma_sqr_right) {
int retval = 0, beta_retval = 0, left_success = 0, right_success = 0;
mpq_t alpha_star_sqr;
mpq_t t_dist_sqr;
mpq_init(alpha_star_sqr);
mpq_init(t_dist_sqr);
mpq_set_ui(alpha_star_sqr, 4, 100);
mpq_mul(alpha_star_sqr, alpha_star_sqr, alpha_star_sqr);
mpq_sub(t_dist_sqr, t_right, t_left); /* t_2 -t_1 */
mpq_mul(t_dist_sqr, t_dist_sqr, t_dist_sqr); /* |t_2 - t_1|^2 */
/* we want either of magic_number_left or _right to be less than alpha_star */
mpq_t magic_number_left;
mpq_t norm_sqr_Jv_left;
mpq_t magic_number_right;
mpq_t norm_sqr_Jv_right;
mpq_init(magic_number_left);
mpq_init(norm_sqr_Jv_left);
mpq_init(magic_number_right);
mpq_init(norm_sqr_Jv_right);
compute_norm_sqr_Jv_rational(&F_left, v, x_left, &norm_sqr_Jv_left);
compute_norm_sqr_Jv_rational(&F_right, v, x_right, &norm_sqr_Jv_right);
/* ||J^-1 * v||^2 */
mpq_set(magic_number_left, norm_sqr_Jv_left);
mpq_set(magic_number_right, norm_sqr_Jv_right);
/* ||J^-1 * v||^2 * g^2 */
mpq_mul(magic_number_left, magic_number_left, gamma_sqr_left);
mpq_mul(magic_number_right, magic_number_right, gamma_sqr_right);
/* ||J^-1 * v||^2 * g^2 * |t_1 - t_2|^2*/
mpq_mul(magic_number_left, magic_number_left, t_dist_sqr);
mpq_mul(magic_number_right, magic_number_right, t_dist_sqr);
/* run the tests */
left_success = lte_sqr_rational(alpha_sqr_left, magic_number_left, alpha_star_sqr);
right_success = lte_sqr_rational(alpha_sqr_right, magic_number_right, alpha_star_sqr);
/* test fails for both left and right */
if (left_success == 0 && right_success == 0)
retval = 0;
else {
rational_complex_number beta_sqr_complex1;
rational_complex_number beta_sqr_complex2;
mpq_t beta_sqr1;
mpq_t beta_sqr2;
mpq_t four;
mpq_t points_dist_sqr;
initialize_rational_number(beta_sqr_complex1);
initialize_rational_number(beta_sqr_complex2);
mpq_init(beta_sqr1);
mpq_init(beta_sqr2);
mpq_init(four);
mpq_init(points_dist_sqr);
mpq_set_ui(four, 4, 1);
/* test succeeds for at least left (possibly right, but use left anyway) */
if (left_success) {
beta_retval = compute_beta_sqr_rational(beta_sqr_complex1, &F_left, x_left);
if (!beta_retval)
mpq_set(beta_sqr1, beta_sqr_complex1->re);
beta_retval = compute_beta_sqr_rational(beta_sqr_complex2, &F_left, x_right);
if (!beta_retval)
mpq_set(beta_sqr2, beta_sqr_complex2->re);
} else { /* test succeeds for right only */
beta_retval = compute_beta_sqr_rational(beta_sqr_complex1, &F_right, x_left);
if (!beta_retval)
mpq_set(beta_sqr1, beta_sqr_complex1->re);
beta_retval = compute_beta_sqr_rational(beta_sqr_complex2, &F_right, x_right);
if (!beta_retval)
mpq_set(beta_sqr2, beta_sqr_complex2->re);
}
/* 4* beta^2 */
mpq_mul(beta_sqr1, four, beta_sqr1);
mpq_mul(beta_sqr2, four, beta_sqr2);
rational_complex_vector points_dist;
initialize_rational_vector(points_dist, x_left->size);
subtract_rational_vector(points_dist, x_left, x_right);
norm_sqr_rational_vector(points_dist_sqr, points_dist); /* ||x_1 - x_2||^2 */
if (lte_sqr_rational(beta_sqr1, beta_sqr2, points_dist_sqr))
retval = 1;
else
retval = 0;
clear_rational_number(beta_sqr_complex1);
clear_rational_number(beta_sqr_complex2);
clear_rational_vector(points_dist);
mpq_clear(four);
mpq_clear(beta_sqr1);
mpq_clear(beta_sqr2);
mpq_clear(points_dist_sqr);
}
mpq_clear(alpha_star_sqr);
mpq_clear(t_dist_sqr);
mpq_clear(magic_number_left);
mpq_clear(magic_number_right);
mpq_clear(norm_sqr_Jv_left);
mpq_clear(norm_sqr_Jv_right);
return retval;
}
/**********************************************
* test for the continuity of a given segment *
**********************************************/
int test_continuity_rational(rational_complex_vector v, mpq_t t_left, mpq_t t_right, rational_complex_vector x_left, rational_complex_vector x_right, polynomial_system F_left, polynomial_system F_right, mpq_t alpha_sqr_left, mpq_t alpha_sqr_right, mpq_t gamma_sqr_left, mpq_t gamma_sqr_right) {
if (is_continuous_rational(v, t_left, t_right, x_left, x_right, F_left, F_right, alpha_sqr_left, alpha_sqr_right, gamma_sqr_left, gamma_sqr_right))
return 1;
else if (is_not_continuous_rational(v, t_left, t_right, x_left, x_right, F_left, F_right, alpha_sqr_left, alpha_sqr_right, gamma_sqr_left, gamma_sqr_right))
return -1;
return 0;
}
/**************************************************************************************
* subdivide a segment (t_1, w_1) <=> (t_2, w_2) for ((t_1 + t_2)/2, (w_1 + w_2) / 2) *
**************************************************************************************/
void subdivide_segment_rational(polynomial_system *base, rational_complex_vector v, mpq_t t_left, mpq_t t_right, rational_complex_vector x_left, rational_complex_vector x_right, mpq_t *t_mid, rational_complex_vector *x_mid, int num_var) {
if (verbosity > BH_VERBOSE)
printf("subdividing... ");
int i, x_left_solution, x_mid_solution, x_right_solution, retval = 0, *rowswaps = NULL;
mpq_t one_half_rational;
mpq_t alpha_sqr;
mpq_t beta_sqr;
mpq_t gamma_sqr;
mpq_t alpha0_sqr;
mpq_init(one_half_rational);
mpq_init(alpha_sqr);
mpq_init(beta_sqr);
mpq_init(gamma_sqr);
mpq_init(alpha0_sqr);
mpq_set_ui(one_half_rational, 1, 2);
mpq_set_ui(alpha0_sqr, 12321, 495616); /* 12321/495616 < ((13 - 3 * sqrt(17))/4)^2 */
rational_complex_number one_half_complex;
initialize_rational_number(one_half_complex);
mpq_set_ui(one_half_complex->re, 1, 2);
mpq_set_ui(one_half_complex->im, 0, 1);
rational_complex_number beta_sqr_complex;
initialize_rational_number(beta_sqr_complex);
mpq_set(beta_sqr_complex->re, beta_sqr);
mpq_set_ui(beta_sqr_complex->im, 0, 1);
rational_complex_matrix LU;
initialize_rational_matrix(LU, 0, 0);
polynomial_system F_left;
polynomial_system F_mid;
polynomial_system F_right;
/* t_left and t_right are fixed, this will not change */
apply_tv_rational(base, &F_left, *t_mid, v);
apply_tv_rational(base, &F_right, *t_mid, v);
rational_complex_vector new_point;
initialize_rational_vector(new_point, num_var);
/* perform a Newton iteration on x_left */
retval = newton_iteration_rational(new_point, beta_sqr_complex, LU, &rowswaps, &F_left, x_left);
if (retval == ERROR_LU_DECOMP) {
fprintf(stderr, "singularity suspected near point ");
print_points_rational(stderr, new_point);
fprintf(stderr, ERROR_MESSAGE);
exit(BH_EXIT_OTHER);
}
copy_rational_vector(x_left, new_point);
x_left_solution = compute_abg_sqr_rational(x_left, &F_left, &alpha_sqr, &beta_sqr, &gamma_sqr);
i = 1;
/* perform Newton iterations on x_left until within tolerance */
while (i < newton_tolerance && mpq_cmp(alpha_sqr, alpha0_sqr) >= 0) {
retval = newton_iteration_rational(new_point, beta_sqr_complex, LU, &rowswaps, &F_left, x_left);
if (retval == ERROR_LU_DECOMP) {
fprintf(stderr, "singularity suspected near point ");
print_points_rational(stderr, new_point);
fprintf(stderr, ERROR_MESSAGE);
exit(BH_EXIT_OTHER);
}
copy_rational_vector(x_left, new_point);
x_left_solution = compute_abg_sqr_rational(x_left, &F_left, &alpha_sqr, &beta_sqr, &gamma_sqr);
i++;
}
/* perform a Newton iteration on x_right */
retval = newton_iteration_rational(new_point, beta_sqr_complex, LU, &rowswaps, &F_right, x_right);
if (retval == ERROR_LU_DECOMP) {
fprintf(stderr, "singularity suspected near point ");
print_points_rational(stderr, new_point);
fprintf(stderr, ERROR_MESSAGE);
exit(BH_EXIT_OTHER);
}
copy_rational_vector(x_right, new_point);
x_right_solution = compute_abg_sqr_rational(x_right, &F_right, &alpha_sqr, &beta_sqr, &gamma_sqr);
i = 1;
/* perform Newton iterations on x_right until within tolerance */
while (i < newton_tolerance && mpq_cmp(alpha_sqr, alpha0_sqr) >= 0) {
retval = newton_iteration_rational(new_point, beta_sqr_complex, LU, &rowswaps, &F_right, x_right);
if (retval == ERROR_LU_DECOMP) {
fprintf(stderr, "singularity suspected near point ");
print_points_rational(stderr, new_point);
fprintf(stderr, ERROR_MESSAGE);
exit(BH_EXIT_OTHER);
}
copy_rational_vector(x_right, new_point);
x_right_solution = compute_abg_sqr_rational(x_right, &F_right, &alpha_sqr, &beta_sqr, &gamma_sqr);
i++;
}
/* t_mid */
mpq_add(*t_mid, t_left, t_right);
mpq_mul(*t_mid, *t_mid, one_half_rational);
/* x_mid */
for (i = 0; i < num_var; i++) {
add_rational((*x_mid)->coord[i], x_left->coord[i], x_right->coord[i]);
multiply_rational((*x_mid)->coord[i], (*x_mid)->coord[i], one_half_complex);
}
/* apply Newton to x_mid */
apply_tv_rational(base, &F_mid, *t_mid, v);
retval = newton_iteration_rational(new_point, beta_sqr_complex, LU, &rowswaps, &F_mid, *x_mid);
if (retval == ERROR_LU_DECOMP) {
fprintf(stderr, "singularity suspected near point ");
print_points_rational(stderr, new_point);
fprintf(stderr, ERROR_MESSAGE);
exit(BH_EXIT_OTHER);
}
copy_rational_vector(*x_mid, new_point);
x_mid_solution = compute_abg_sqr_rational(*x_mid, &F_right, &alpha_sqr, &beta_sqr, &gamma_sqr);
i = 1;
/* perform Newton iterations on x_mid until within tolerance */
while (i < newton_tolerance && mpq_cmp(alpha_sqr, alpha0_sqr) >= 0) {
retval = newton_iteration_rational(new_point, beta_sqr_complex, LU, &rowswaps, &F_right, *x_mid);
if (retval == ERROR_LU_DECOMP) {
fprintf(stderr, "singularity suspected near point ");
print_points_rational(stderr, new_point);
fprintf(stderr, ERROR_MESSAGE);
exit(BH_EXIT_OTHER);
}
copy_rational_vector(*x_mid, new_point);
x_mid_solution = compute_abg_sqr_rational(*x_mid, &F_right, &alpha_sqr, &beta_sqr, &gamma_sqr);
i++;
}
if (verbosity > BH_VERBOSE)
gmp_printf("new intervals are [%Qd, %Qd] and [%Qd, %Qd]\n", t_left, *t_mid, *t_mid, t_right);
mpq_clear(one_half_rational);
mpq_clear(alpha_sqr);
mpq_clear(beta_sqr);
mpq_clear(gamma_sqr);
mpq_clear(alpha0_sqr);
clear_rational_number(one_half_complex);
clear_rational_number(beta_sqr_complex);
clear_rational_matrix(LU);
clear_polynomial_system(&F_mid);
clear_rational_vector(new_point);
free(rowswaps);
rowswaps = NULL;
}
/*******************************
* apply f + tv for a single t *
*******************************/
void apply_tv_rational(polynomial_system *base, polynomial_system *F, mpq_t t, rational_complex_vector v) {
int i, j, k, num_terms, num_var, num_poly;
/* get information from base */
num_var = base->numVariables;
num_poly = base->numPolynomials;
errno = 0;
F->numVariables = num_var;
F->numPolynomials = num_poly;
F->maximumDegree = base->maximumDegree;
F->isReal = 0;
F->numExponentials = 0;
F->polynomials = malloc(num_var * sizeof(polynomial));
if (F->polynomials == NULL) {
snprintf(error_string, (size_t) termwidth, "Couldn't alloc: %s\n", strerror(errno));
print_error(error_string, stderr);
exit(BH_EXIT_MEMORY);
}
F->exponentials = NULL;
mpq_init(F->norm_sqr);
/* for each polynomial in F */
for (i = 0; i < num_var; i++) {
num_terms = base->polynomials[i].numTerms + 1;
/* copy base->p[i] */
polynomial p;
p.numTerms = num_terms;
p.numVariables = num_var;
p.degree = base->polynomials[i].degree;
/* copy exponents from base->polynomials[i] */
errno = 0;
p.exponents = malloc(num_terms * sizeof(int *));
if (p.exponents == NULL) {
snprintf(error_string, (size_t) termwidth, "Couldn't alloc: %s\n", strerror(errno));
print_error(error_string, stderr);
exit(BH_EXIT_MEMORY);
}
for (j = 0; j < num_terms - 1; j++) {
p.exponents[j] = malloc(num_var * sizeof(int));
if (p.exponents[j] == NULL) {
snprintf(error_string, (size_t) termwidth, "Couldn't alloc: %s\n", strerror(errno));
print_error(error_string, stderr);
exit(BH_EXIT_MEMORY);
}
for (k = 0; k < num_var; k++)
p.exponents[j][k] = base->polynomials[i].exponents[j][k];
}
/* add 0's for exponents in last (constant) term */
p.exponents[j] = malloc(num_var * sizeof(int));
if (p.exponents[j] == NULL) {
snprintf(error_string, (size_t) termwidth, "Couldn't alloc: %s\n", strerror(errno));
print_error(error_string, stderr);
exit(BH_EXIT_MEMORY);
}
for (k = 0; k < num_var; k++)
p.exponents[j][k] = 0;
/* copy the coefficients */
p.coeff = malloc(num_terms * sizeof(rational_complex_number));
if (p.coeff == NULL) {
snprintf(error_string, (size_t) termwidth, "Couldn't alloc: %s\n", strerror(errno));
print_error(error_string, stderr);
exit(BH_EXIT_MEMORY);
}
for (j = 0; j < num_terms - 1; j++) {
initialize_rational_number(p.coeff[j]);
set_rational_number(p.coeff[j], base->polynomials[i].coeff[j]);
}
/* now add tv */
initialize_rational_number(p.coeff[j]);
mpq_mul(p.coeff[j]->re, t, v->coord[i]->re);
mpq_mul(p.coeff[j]->im, t, v->coord[i]->im);
mpq_init(p.norm_sqr);
norm_sqr_polynomial(p.norm_sqr, &p);
F->polynomials[i] = p;
}
}
/***************************************
* get alpha^2, beta^2, gamma^2 values *
***************************************/
int compute_abg_sqr_rational(rational_complex_vector points, polynomial_system *F, mpq_t *alpha_sqr, mpq_t *beta_sqr, mpq_t *gamma_sqr) {
int res = 0, retval = 0, num_approx_solns = 0, num_var = F->numVariables;
rational_point_struct P;
/* setup point struct and determine if it is an approximate solution */
initialize_rational_point_struct(&P, num_var);
/* set to active */
P.isActive = 1;
/* copy rational point */
copy_rational_vector(P.origX, points);
copy_rational_vector(P.x, points);
/* compute ||x||_2^2 */
norm_sqr_rational_vector(P.norm_sqr_x, P.x);
/* compute alpha^2, beta^2, & gamma^2 */
res = compute_alpha_beta_gamma_sqr_rational(P.Nx, P.alpha_sqr, P.beta_sqr, P.gamma_sqr, F, P.x);
mpq_set(*alpha_sqr, P.alpha_sqr->re);
mpq_set(*beta_sqr, P.beta_sqr->re);
mpq_set(*gamma_sqr, P.gamma_sqr->re);
//copy_rational_vector(points, P.Nx);
/* check to see if we have successfully computed alpha_sqr, beta_sqr, & gamma_sqr */
if (res == EXACT_SOLUTION_LU_ERROR) {
/* exact solution */
retval = 1;
} else if (res) {
/* unknown */
retval = 0;
} else {
/* determine if alpha_sqr is small enough to be an approximate solution */
num_approx_solns += P.isApproxSoln = determine_approximate_solution_rational(P.alpha_sqr);
retval = num_approx_solns;
}
clear_rational_point_struct(&P);
return retval;
}
/***********************************************************************************
* test that each w is in the quadratic convergence basin and check for continuity *
***********************************************************************************/
void test_pairwise_rational(polynomial_system *system, rational_complex_vector *v, mpq_t t_left, mpq_t t_right, rational_complex_vector x_left, rational_complex_vector x_right, int num_var, int iter, mpq_t **t_final, rational_complex_vector **x_final, rational_complex_vector **sing, int *tested, int *succeeded, int *failed, int *num_sing, int check_left) {
int x_left_solution, x_right_solution, seg_continuous;
polynomial_system F_left, F_right;
mpq_t alpha_sqr_left, beta_sqr_left, gamma_sqr_left, alpha_sqr_right, beta_sqr_right, gamma_sqr_right, beta_sqr_min;
mpq_init(alpha_sqr_left);
mpq_init(beta_sqr_left);
mpq_init(gamma_sqr_left);
mpq_init(alpha_sqr_right);
mpq_init(beta_sqr_right);
mpq_init(gamma_sqr_right);
mpq_init(beta_sqr_min);
apply_tv_rational(system, &F_left, t_left, *v);
apply_tv_rational(system, &F_right, t_right, *v);
x_left_solution = compute_abg_sqr_rational(x_left, &F_left, &alpha_sqr_left, &beta_sqr_left, &gamma_sqr_left);
x_right_solution = compute_abg_sqr_rational(x_right, &F_right, &alpha_sqr_right, &beta_sqr_right, &gamma_sqr_right);
/* test if we've been through too many times */
if (iter > subd_tolerance) {
*tested += 1;
if (verbosity > BH_LACONIC)
gmp_printf("unsure whether segment [%Qd, %Qd] is continuous\n", t_left, t_right);
fprint_uncertain_rational(t_left, t_right, x_left, x_right, alpha_sqr_left, alpha_sqr_right, beta_sqr_left, beta_sqr_right, gamma_sqr_left, gamma_sqr_right);
/* alert user to singularities */
mpz_t denom;
mpz_init(denom);
if (check_left) {
mpq_get_den(denom, gamma_sqr_left);
if (mpz_cmp_ui(denom, 0) == 0) {
gmp_fprintf(stderr, "singularity found for t = %Qd at point ", t_left);
print_points_rational(stderr, x_left);
}
}
mpq_get_den(denom, gamma_sqr_right);
if (mpz_cmp_ui(denom, 0) == 0) {
gmp_fprintf(stderr, "singularity found for t = %Qd at point ", t_right);
print_points_rational(stderr, x_right);
}
mpz_clear(denom);
return;
}
mpq_set_min(beta_sqr_min, beta_sqr_left, beta_sqr_right);
/* perform Newton iterations on non-compliant points */
int newton_counter = 1;
while (!x_left_solution || !x_right_solution) {
if (newton_counter == newton_tolerance) {
char err_string[] = "Points not in convergence basin; aborting";
print_error(err_string, stderr);
exit(BH_EXIT_OTHER);
}
int retval, *rowswaps = NULL;
rational_complex_number beta_sqr_complex;
rational_complex_vector new_point;
rational_complex_matrix LU;
/* perform Newton iterations if x_left is not a solution */
if (!x_left_solution) {
if (verbosity > BH_VERBOSE) {
if (newton_counter % 10 == 1 && newton_counter % 100 != 11)
fprintf(stderr, "performing %dst Newton iteration on x_left", newton_counter);
else if (newton_counter % 10 == 2 && newton_counter % 100 != 12)
fprintf(stderr, "performing %dnd Newton iteration on x_left", newton_counter);
else if (newton_counter % 10 == 3 && newton_counter % 100 != 13)
fprintf(stderr, "performing %drd Newton iteration on x_left", newton_counter);
else
fprintf(stderr, "performing %dth Newton iteration on x_left", newton_counter);
gmp_fprintf(stderr, " (interval: [%Qd, %Qd])\n", t_left, t_right);
}
initialize_rational_number(beta_sqr_complex);
initialize_rational_vector(new_point, num_var);
initialize_rational_matrix(LU, 0, 0);
retval = newton_iteration_rational(new_point, beta_sqr_complex, LU, &rowswaps, &F_left, x_left);
if (retval == ERROR_LU_DECOMP) {
fprintf(stderr, "singularity suspected near point ");
print_points_rational(stderr, new_point);
fprintf(stderr, ERROR_MESSAGE);
exit(BH_EXIT_OTHER);
}
copy_rational_vector(x_left, new_point);
x_left_solution = compute_abg_sqr_rational(x_left, &F_left, &alpha_sqr_left, &beta_sqr_left, &gamma_sqr_left);
clear_rational_number(beta_sqr_complex);
clear_rational_vector(new_point);
clear_rational_matrix(LU);
free(rowswaps);
rowswaps = NULL;
}
/* perform Newton iterations if x_right is not a solution */
if (!x_right_solution) {
if (verbosity > BH_VERBOSE) {
if (newton_counter % 10 == 1 && newton_counter % 100 != 11)
fprintf(stderr, "performing %dst Newton iteration on x_right", newton_counter);
else if (newton_counter % 10 == 2 && newton_counter % 100 != 12)
fprintf(stderr, "performing %dnd Newton iteration on x_right", newton_counter);
else if (newton_counter % 10 == 3 && newton_counter % 100 != 13)
fprintf(stderr, "performing %drd Newton iteration on x_right", newton_counter);
else
fprintf(stderr, "performing %dth Newton iteration on x_right", newton_counter);
gmp_fprintf(stderr, " (interval: [%Qd, %Qd])\n", t_left, t_right);
}
initialize_rational_number(beta_sqr_complex);
initialize_rational_vector(new_point, num_var);
initialize_rational_matrix(LU, 0, 0);
retval = newton_iteration_rational(new_point, beta_sqr_complex, LU, &rowswaps, &F_right, x_right);
if (retval == ERROR_LU_DECOMP) {
fprintf(stderr, "singularity suspected near point ");
print_points_rational(stderr, new_point);
fprintf(stderr, ERROR_MESSAGE);
exit(BH_EXIT_OTHER);
}
copy_rational_vector(x_right, new_point);
x_right_solution = compute_abg_sqr_rational(x_right, &F_right, &alpha_sqr_right, &beta_sqr_right, &gamma_sqr_right);
clear_rational_number(beta_sqr_complex);
clear_rational_vector(new_point);
clear_rational_matrix(LU);
free(rowswaps);
rowswaps = NULL;
}
newton_counter++;
}
if (verbosity > BH_CHATTY)
gmp_printf("testing segment [%Qd, %Qd]\n", t_left, t_right);
seg_continuous = test_continuity_rational(*v, t_left, t_right, x_left, x_right, F_left, F_right, alpha_sqr_left, alpha_sqr_right, gamma_sqr_left, gamma_sqr_right);
if (seg_continuous == 0) {
mpq_t t_mid;
mpq_init(t_mid);
rational_complex_vector x_mid;
initialize_rational_vector(x_mid, num_var);
subdivide_segment_rational(system, *v, t_left, t_right, x_left, x_right, &t_mid, &x_mid, num_var);
/* recurse! */
test_pairwise_rational(system, v, t_left, t_mid, x_left, x_mid, num_var, iter + 1, t_final, x_final, sing, tested, succeeded, failed, num_sing, 0);
test_pairwise_rational(system, v, t_mid, t_right, x_mid, x_right, num_var, iter + 1, t_final, x_final, sing, tested, succeeded, failed, num_sing, 0);
mpq_clear(t_mid);
clear_rational_vector(x_mid);
} else if (seg_continuous == 1) {
if (verbosity > BH_LACONIC)
gmp_printf("segment [%Qd, %Qd] is continuous\n", t_left, t_right);
fprint_continuous_rational(t_left, t_right, x_left, x_right, alpha_sqr_left, alpha_sqr_right, beta_sqr_left, beta_sqr_right, gamma_sqr_left, gamma_sqr_right);
*tested += 1;
*succeeded += 1;
/* alert user to singularities */
mpz_t denom;
mpz_init(denom);
if (check_left) {
mpq_get_den(denom, gamma_sqr_left);
if (mpz_cmp_ui(denom, 0) == 0) {
gmp_fprintf(stderr, "singularity found for t = %Qd at point ", t_left);
print_points_rational(stderr, x_left);
}
}
mpq_get_den(denom, gamma_sqr_right);
if (mpz_cmp_ui(denom, 0) == 0) {
gmp_fprintf(stderr, "singularity found for t = %Qd at point ", t_right);
print_points_rational(stderr, x_right);
}
mpz_clear(denom);
/* copy t and w values into t_final and x_final respectively */
errno = 0;
mpq_t *t_final_tmp = realloc(*t_final, (size_t) (*succeeded + 1) * sizeof(mpq_t));
if (t_final_tmp == NULL) {
snprintf(error_string, (size_t) termwidth, "Couldn't realloc: %s\n", strerror(errno));
print_error(error_string, stderr);
exit(BH_EXIT_MEMORY);
} else {
*t_final = t_final_tmp;
mpq_init((*t_final)[*succeeded]);
mpq_set((*t_final)[*succeeded], t_right);
}
errno = 0;
rational_complex_vector *x_final_tmp = realloc(*x_final, (size_t) (*succeeded + 1) * sizeof(rational_complex_vector));
if (x_final_tmp == NULL) {
snprintf(error_string, (size_t) termwidth, "Couldn't realloc: %s\n", strerror(errno));
print_error(error_string, stderr);
exit(BH_EXIT_MEMORY);
} else {
*x_final = x_final_tmp;
initialize_rational_vector((*x_final)[*succeeded], x_right->size);
copy_rational_vector((*x_final)[*succeeded], x_right);
}
} else {
if (verbosity > BH_LACONIC)
gmp_printf("segment [%Qd, %Qd] is not continuous\n", t_left, t_right);
fprint_discontinuous_rational(t_left, t_right, x_left, x_right, alpha_sqr_left, alpha_sqr_right, beta_sqr_left, beta_sqr_right, gamma_sqr_left, gamma_sqr_right);
/* alert user to singularities */
mpz_t denom;
mpz_init(denom);
if (check_left) {
mpq_get_den(denom, gamma_sqr_left);
if (mpz_cmp_ui(denom, 0) == 0) {
gmp_fprintf(stderr, "singularity found for t = %Qd at point ", t_left);
print_points_rational(stderr, x_left);
}
}
mpq_get_den(denom, gamma_sqr_right);
if (mpz_cmp_ui(denom, 0) == 0) {
gmp_fprintf(stderr, "singularity found for t = %Qd at point ", t_right);
print_points_rational(stderr, x_right);
}
mpz_clear(denom);
*tested += 1;
*failed += 1;
}
/* free up stuff */
clear_polynomial_system(&F_left);
clear_polynomial_system(&F_right);
mpq_clear(alpha_sqr_left);
mpq_clear(beta_sqr_left);
mpq_clear(gamma_sqr_left);
mpq_clear(alpha_sqr_right);
mpq_clear(beta_sqr_right);
mpq_clear(gamma_sqr_right);
mpq_clear(beta_sqr_min);
}
/*******************
* certify H(x, t) *
*******************/
void test_system_rational(polynomial_system *system, void *v, void *t, void *x, int num_points, void **t_final, void **x_final, void **sing, int *tested, int *succeeded, int *failed, int *num_sing) {
int i, num_var = system->numVariables;
rational_complex_vector *v_rational = (rational_complex_vector *) v;
mpq_t *t_rational = (mpq_t *) t;
rational_complex_vector *x_rational = (rational_complex_vector *) x;
mpq_t **t_final_rational = (mpq_t **) t_final;
rational_complex_vector **x_final_rational = (rational_complex_vector **) x_final;
rational_complex_vector **sing_rational = (rational_complex_vector **) sing;
/* set up t_final and x_final */
*t_final_rational = malloc(sizeof(mpq_t));
if (*t_final_rational == NULL) {
snprintf(error_string, (size_t) termwidth, "Couldn't alloc: %s\n", strerror(errno));
print_error(error_string, stderr);
exit(BH_EXIT_MEMORY);
}
mpq_init((*t_final_rational)[0]);
mpq_set((*t_final_rational)[0], t_rational[0]);
*x_final_rational = malloc(sizeof(rational_complex_vector));
if (*x_final_rational == NULL) {
snprintf(error_string, (size_t) termwidth, "Couldn't alloc: %s\n", strerror(errno));
print_error(error_string, stderr);
exit(BH_EXIT_MEMORY);
}
initialize_rational_vector((*x_final_rational)[0], x_rational[0]->size);
copy_rational_vector((*x_final_rational)[0], x_rational[0]);
/* for each t_i, t_{i+1} */
/* this is the part that needs to get parallelized */
for (i = 0; i < num_points - 1; i++) {
test_pairwise_rational(system, v_rational, t_rational[i], t_rational[i+1], x_rational[i], x_rational[i+1], num_var, 1, t_final_rational, x_final_rational, sing_rational, tested, succeeded, failed, num_sing, (i == 0));
}
}